Abstract

The transformation by a lens of a polychromatic laser beam composed of on-axis superposed monochromatic TEM00 Gaussian modes in the paraxial approximation is studied. The chromatic aberrations are described by allowing the waist position on the z axis and the Rayleigh range to depend on wavelength. The beam radius, the far-field divergence, the Rayleigh range, the beam product, the beam propagation factor, and the kurtosis parameter are calculated. The relationship between the fourth-order and the second-order moments of Hermite–Gaussian and Laguerre–Gaussian modes is obtained and is used for calculating kurtosis parameter. The results are generalized to polychromatic modes of higher orders. It is shown that the on-axis superposition of monochromatic TEM00 modes with no chromatic aberration is leptokurtic.

© 2001 Optical Society of America

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References

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  1. R. Simon, N. Mukunda, G. C. E. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
    [CrossRef]
  2. L. Shimon, R. Prochaska, E. Keren, “Generalized beam parameters and transformation laws for partially coherent light,” Appl. Opt. 27, 3696–3703 (1988).
    [CrossRef]
  3. J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
    [CrossRef]
  4. Q. Lin, S. Wang, J. Alda, E. Bernabeu, “Transformation of pulsed nonideal beams in a four-dimension domain,” Opt. Lett. 18, 669–671 (1993).
    [CrossRef] [PubMed]
  5. P. M. Mejı́as, R. Martı́nez-Herrero, “Time-resolved spatial parametric characterization of pulsed light beams,” Opt. Lett. 20, 660–662 (1995).
    [CrossRef] [PubMed]
  6. Q. Cao, D. Ximing, “Spatial parametric characterization of general polychromatic beams,” Opt. Commun. 142, 135–145 (1997).
    [CrossRef]
  7. C. J. R. Sheppard, X. Gan, “Free-space propagation of femtosecond light pulses,” Opt. Commun. 133, 1–6 (1997).
    [CrossRef]
  8. M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
    [CrossRef]
  9. O. E. Martı́nez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988).
    [CrossRef]
  10. O. E. Martı́nez, “Matrix formalism for dispersive laser cavities,” IEEE J. Quantum Electron. 25, 296–300 (1989).
    [CrossRef]
  11. A. G. Kostenbauder, “Ray-pulse matrices: a rotational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
    [CrossRef]
  12. M. A. Porras, “Propagation of single-cycle pulsed light beams in dispersive media,” Phys. Rev. A 60, 5069–5073 (1999).
    [CrossRef]
  13. L. Martı́-López, O. Mendoza-Yero, “Effect of chromatic aberration on Gaussian beams: non-dispersive laser resonators,” Opt. Laser Technol. 31, 239–245 (1999).
    [CrossRef]
  14. A. E. Siegman, “Defining, measuring, and optimizing laser beam quality,” in Laser Resonators and Coherent Optics: Modeling, Technology and Applications, A. Bhowmik, ed., Proc. SPIE1868, 2–12 (1993).
    [CrossRef]
  15. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), p. 1267.
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  18. M. Y. Klimkov, Osnovy Rasschiota Optiko-Elektronikh Priborov s Lazerami (Fundamentals of Calculus of Optical and Electronic Devices with Lasers) (Sovietskoye Radio, Moscow, 1978), p. 264.
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    [CrossRef] [PubMed]
  20. H. Kogelnik, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  21. L. Marti-López, O. Mendoza-Yero, “Polychromatic Gaussian beams emitted by dispersive laser resonators,” Opt. Laser Technol. 33, 1–5 (2001).
    [CrossRef]
  22. R. D. Bock, Multivariate Statistical Methods in Behavioral Research (McGraw-Hill, New York, 1975), p. 623.
  23. G. Piquero, P. M. Mejı́as, R. Martı́nez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
    [CrossRef]
  24. R. Martı́nez-Herrero, G. Piquero, P. M. Mejı́as, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
    [CrossRef]
  25. S. Saghafi, C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
    [CrossRef]
  26. U. Vokinger, R. Dändliker, P. Blattner, H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. 157, 218–224 (1998).
    [CrossRef]

2001 (1)

L. Marti-López, O. Mendoza-Yero, “Polychromatic Gaussian beams emitted by dispersive laser resonators,” Opt. Laser Technol. 33, 1–5 (2001).
[CrossRef]

1999 (2)

M. A. Porras, “Propagation of single-cycle pulsed light beams in dispersive media,” Phys. Rev. A 60, 5069–5073 (1999).
[CrossRef]

L. Martı́-López, O. Mendoza-Yero, “Effect of chromatic aberration on Gaussian beams: non-dispersive laser resonators,” Opt. Laser Technol. 31, 239–245 (1999).
[CrossRef]

1998 (3)

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[CrossRef]

S. Saghafi, C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
[CrossRef]

U. Vokinger, R. Dändliker, P. Blattner, H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. 157, 218–224 (1998).
[CrossRef]

1997 (2)

Q. Cao, D. Ximing, “Spatial parametric characterization of general polychromatic beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

C. J. R. Sheppard, X. Gan, “Free-space propagation of femtosecond light pulses,” Opt. Commun. 133, 1–6 (1997).
[CrossRef]

1995 (2)

R. Martı́nez-Herrero, G. Piquero, P. M. Mejı́as, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[CrossRef]

P. M. Mejı́as, R. Martı́nez-Herrero, “Time-resolved spatial parametric characterization of pulsed light beams,” Opt. Lett. 20, 660–662 (1995).
[CrossRef] [PubMed]

1994 (1)

G. Piquero, P. M. Mejı́as, R. Martı́nez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
[CrossRef]

1993 (1)

1991 (1)

1990 (1)

A. G. Kostenbauder, “Ray-pulse matrices: a rotational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[CrossRef]

1989 (1)

O. E. Martı́nez, “Matrix formalism for dispersive laser cavities,” IEEE J. Quantum Electron. 25, 296–300 (1989).
[CrossRef]

1988 (3)

O. E. Martı́nez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988).
[CrossRef]

R. Simon, N. Mukunda, G. C. E. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

L. Shimon, R. Prochaska, E. Keren, “Generalized beam parameters and transformation laws for partially coherent light,” Appl. Opt. 27, 3696–3703 (1988).
[CrossRef]

1983 (1)

1981 (1)

1980 (1)

1966 (1)

H. Kogelnik, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Alda, J.

Bernabeu, E.

Blattner, P.

U. Vokinger, R. Dändliker, P. Blattner, H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. 157, 218–224 (1998).
[CrossRef]

Bock, R. D.

R. D. Bock, Multivariate Statistical Methods in Behavioral Research (McGraw-Hill, New York, 1975), p. 623.

Cao, Q.

Q. Cao, D. Ximing, “Spatial parametric characterization of general polychromatic beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

Carter, W. H.

Dändliker, R.

U. Vokinger, R. Dändliker, P. Blattner, H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. 157, 218–224 (1998).
[CrossRef]

Gan, X.

C. J. R. Sheppard, X. Gan, “Free-space propagation of femtosecond light pulses,” Opt. Commun. 133, 1–6 (1997).
[CrossRef]

Herzig, H. P.

U. Vokinger, R. Dändliker, P. Blattner, H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. 157, 218–224 (1998).
[CrossRef]

Iffländer, R.

Keren, E.

Klimkov, M. Y.

M. Y. Klimkov, Osnovy Rasschiota Optiko-Elektronikh Priborov s Lazerami (Fundamentals of Calculus of Optical and Electronic Devices with Lasers) (Sovietskoye Radio, Moscow, 1978), p. 264.

Kogelnik, H.

H. Kogelnik, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Kortz, H. P.

Kostenbauder, A. G.

A. G. Kostenbauder, “Ray-pulse matrices: a rotational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[CrossRef]

Lin, Q.

Marti´-López, L.

L. Martı́-López, O. Mendoza-Yero, “Effect of chromatic aberration on Gaussian beams: non-dispersive laser resonators,” Opt. Laser Technol. 31, 239–245 (1999).
[CrossRef]

Marti´nez, O. E.

O. E. Martı́nez, “Matrix formalism for dispersive laser cavities,” IEEE J. Quantum Electron. 25, 296–300 (1989).
[CrossRef]

O. E. Martı́nez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988).
[CrossRef]

Marti´nez-Herrero, R.

P. M. Mejı́as, R. Martı́nez-Herrero, “Time-resolved spatial parametric characterization of pulsed light beams,” Opt. Lett. 20, 660–662 (1995).
[CrossRef] [PubMed]

R. Martı́nez-Herrero, G. Piquero, P. M. Mejı́as, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[CrossRef]

G. Piquero, P. M. Mejı́as, R. Martı́nez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
[CrossRef]

J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

Marti-López, L.

L. Marti-López, O. Mendoza-Yero, “Polychromatic Gaussian beams emitted by dispersive laser resonators,” Opt. Laser Technol. 33, 1–5 (2001).
[CrossRef]

Meji´as, P. M.

P. M. Mejı́as, R. Martı́nez-Herrero, “Time-resolved spatial parametric characterization of pulsed light beams,” Opt. Lett. 20, 660–662 (1995).
[CrossRef] [PubMed]

R. Martı́nez-Herrero, G. Piquero, P. M. Mejı́as, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[CrossRef]

G. Piquero, P. M. Mejı́as, R. Martı́nez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
[CrossRef]

J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

Mendoza-Yero, O.

L. Marti-López, O. Mendoza-Yero, “Polychromatic Gaussian beams emitted by dispersive laser resonators,” Opt. Laser Technol. 33, 1–5 (2001).
[CrossRef]

L. Martı́-López, O. Mendoza-Yero, “Effect of chromatic aberration on Gaussian beams: non-dispersive laser resonators,” Opt. Laser Technol. 31, 239–245 (1999).
[CrossRef]

Mukunda, N.

R. Simon, N. Mukunda, G. C. E. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

Piquero, G.

R. Martı́nez-Herrero, G. Piquero, P. M. Mejı́as, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[CrossRef]

G. Piquero, P. M. Mejı́as, R. Martı́nez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
[CrossRef]

Porras, M. A.

M. A. Porras, “Propagation of single-cycle pulsed light beams in dispersive media,” Phys. Rev. A 60, 5069–5073 (1999).
[CrossRef]

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[CrossRef]

Prochaska, R.

Saghafi, S.

S. Saghafi, C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
[CrossRef]

Self, S. A.

Serna, J.

Sheppard, C. J. R.

S. Saghafi, C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
[CrossRef]

C. J. R. Sheppard, X. Gan, “Free-space propagation of femtosecond light pulses,” Opt. Commun. 133, 1–6 (1997).
[CrossRef]

Shimon, L.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), p. 1267.

A. E. Siegman, “Defining, measuring, and optimizing laser beam quality,” in Laser Resonators and Coherent Optics: Modeling, Technology and Applications, A. Bhowmik, ed., Proc. SPIE1868, 2–12 (1993).
[CrossRef]

Simon, R.

R. Simon, N. Mukunda, G. C. E. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

Sudarshan, G. C. E.

R. Simon, N. Mukunda, G. C. E. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

Vokinger, U.

U. Vokinger, R. Dändliker, P. Blattner, H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. 157, 218–224 (1998).
[CrossRef]

Wang, S.

Weber, H.

Ximing, D.

Q. Cao, D. Ximing, “Spatial parametric characterization of general polychromatic beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

Appl. Opt. (4)

IEEE J. Quantum Electron. (3)

O. E. Martı́nez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988).
[CrossRef]

O. E. Martı́nez, “Matrix formalism for dispersive laser cavities,” IEEE J. Quantum Electron. 25, 296–300 (1989).
[CrossRef]

A. G. Kostenbauder, “Ray-pulse matrices: a rotational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (7)

Q. Cao, D. Ximing, “Spatial parametric characterization of general polychromatic beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

C. J. R. Sheppard, X. Gan, “Free-space propagation of femtosecond light pulses,” Opt. Commun. 133, 1–6 (1997).
[CrossRef]

R. Simon, N. Mukunda, G. C. E. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

G. Piquero, P. M. Mejı́as, R. Martı́nez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
[CrossRef]

R. Martı́nez-Herrero, G. Piquero, P. M. Mejı́as, “On the propagation of the kurtosis parameter of general beams,” Opt. Commun. 115, 225–232 (1995).
[CrossRef]

S. Saghafi, C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
[CrossRef]

U. Vokinger, R. Dändliker, P. Blattner, H. P. Herzig, “Unconventional treatment of focal shift,” Opt. Commun. 157, 218–224 (1998).
[CrossRef]

Opt. Laser Technol. (2)

L. Marti-López, O. Mendoza-Yero, “Polychromatic Gaussian beams emitted by dispersive laser resonators,” Opt. Laser Technol. 33, 1–5 (2001).
[CrossRef]

L. Martı́-López, O. Mendoza-Yero, “Effect of chromatic aberration on Gaussian beams: non-dispersive laser resonators,” Opt. Laser Technol. 31, 239–245 (1999).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

M. A. Porras, “Propagation of single-cycle pulsed light beams in dispersive media,” Phys. Rev. A 60, 5069–5073 (1999).
[CrossRef]

Phys. Rev. E (1)

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[CrossRef]

Proc. IEEE (1)

H. Kogelnik, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Other (4)

R. D. Bock, Multivariate Statistical Methods in Behavioral Research (McGraw-Hill, New York, 1975), p. 623.

M. Y. Klimkov, Osnovy Rasschiota Optiko-Elektronikh Priborov s Lazerami (Fundamentals of Calculus of Optical and Electronic Devices with Lasers) (Sovietskoye Radio, Moscow, 1978), p. 264.

A. E. Siegman, “Defining, measuring, and optimizing laser beam quality,” in Laser Resonators and Coherent Optics: Modeling, Technology and Applications, A. Bhowmik, ed., Proc. SPIE1868, 2–12 (1993).
[CrossRef]

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), p. 1267.

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Figures (2)

Fig. 1
Fig. 1

Stable laser resonator, composed of a dispersive lenslike active medium, a spherical mirror, and a flat mirror. The substrate of the flat mirror is regarded as a dispersive plane-parallel plate. zw, zR, and σw denote the waist position, the Rayleigh range, and the waist radius, respectively, of a monochromatic component of the beam with wavelength λ.

Fig. 2
Fig. 2

Monochromatic TEM00 beam transformed by a thin lens. zw, zws are the waist positions of the input beam and of the output beam, respectively; zR, zRs are the Rayleigh ranges of the input beam and of the output beam, respectively, and σw, σws are the waist radius of the input beam and of the output beam, respectively.

Tables (2)

Tables Icon

Table 1 Values of ϵmn/γmn2 for Hermite–Gaussian Modes

Tables Icon

Table 2 Values of ϵmn/γmn2 for Laguerre–Gaussian Modes

Equations (96)

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2u˜(x, y, z, λ)x2+2u˜(x, y, z, λ)y2-4πjλ u˜(x, y, z, λ)z=0
I(x, y, z, λ)=ρ(λ)|u(x, y, z, λ)|2,
--|u(x, y, z, λ)|2dxdy=1.
I(x, y, z)=0I(x, y, z, λ)dλ,
ρ(λ)=--I(x, y, z, λ)dxdy,
P=--0I(x, y, z, λ)dxdydλ.
σP2(z)1P --r2I(x, y, z)dxdy,
σ2(z, λ)--r2|u(x, y, z, λ)|2dxdy.
zws=f-β(zw+f ),
σws2=βσw2,
σw2=λzR2π,
zRs=βzR,
θs2=β-1θ2
θ2=λ2πzR,
β=1+2zwf+zw2+zR2f2-1.
σs2=σws21+(z-zws)2zRs2.
σs2=λ2πzR zw2+zR2+z2-2zzw+2 z2zw-z(zw2+zR2)f+z2(zw2+zR2)f2.
σPs2(z)=1P 0ρ(λ)σs2dλ=σw2¯+2G0zw¯+α0zw¯2+σ12+2G1z+α1z2,
σw2¯=12πP 0ρ(λ)λzRdλ,
zw¯=1P 0ρ(λ)zwdλ,
σ12=12πP 0ρ(λ) λδ2zR dλ;δ=zw-zw¯,
G0=12πP 0ρ(λ) λδzR dλ,
gj=12πP 0ρ(λ) λfβjzR dλ,j=0, 1,
G1=G0+zw¯α0+g0-g1
αj=12πP 0ρ(λ) λβjzR dλ=1P 0ρ(λ) θ2βj dλ,j=0, 1.
θ2¯=α0,
θs2¯=α1
zPws=-G1α1=g1-g0θs2¯-θ2¯zPwθs2¯,
zPw=zw¯+G0/θ2¯.
WDD=(zPw-zw)θ2¯.
WDD=-WDDs.
σPws2=σPw2+θ2¯zPw2-θs2¯zPws2,
σPw2=σw2¯+σ12-G02/θ2¯.
σPs2(z)=σPws2+θs2¯(z-zPws)2.
σPws2=βPoσPw2,
βPo=1+zPw2θ2¯σPw2-θ2¯2θs2¯ zPwσPw-g1-g0σPwθ2¯2.
σPws21+θs2¯zPws2σPws2=σPw21+θ2¯zPw2σPw2.
zRP2=σPw2/θ2¯;zRPs2=σPws2/θs2¯.
zRPs2=βPo(θ2¯/θs2¯)zRP2.
BPPs=σPwsθP,
MPs2=2π BPPsλ¯,
BPPs=BPP[(θ2/β)¯(βPo/θ2¯)]1/2
MPs2=MP2[(θ2/β)¯(βPo/θ2¯)]1/2
σwsθs=σwθ=λ/2π.
σwsθs¯=σwθ¯=λ¯/2π.
(σwsθs¯)/λ¯s=(σwθ¯)/λ¯.
K=σP4/(σP2)2,
σP4=1P --r4I(x, y, z)dxdy.
K=1P 0ρ(λ)σ4(z, λ)dλ1P 0ρ(λ)σ2(z, λ)dλ2,
σ4(z, λ)--r4|u(x, y, z, λ)|2dxdy.
σ2(z, λ)=w2(z, λ)2=σw21+(z-zw)2zR2,
σ4(z, λ)=w4(z, λ)2=2σw41+(z-zw)2zR22,
K=2 1P 0ρ(λ)σw41+(z-zw)2zR22dλ1P 0ρ(λ)σw21+(z-zw)2zR2dλ2.
K=2+2(λ-λ¯)2(λ¯)2¯.
dKdz=-8Kw2(z, λ)¯(z-zw)σw2zR2¯+16[w2(z, λ)¯]2(z-zw)σw2w2(z, λ)zR2¯.
Knearfield2 λ2zR2¯(λzR¯)2.
Kfarfield=2 λ2zR-2¯(λzR-1¯)2.
Knearfield2 λ2zRs2¯(λzRs¯)2=2 λ2β2zR2¯(λβzR¯)2
Kfarfield=2 λ2zRs-2¯(λzRs-1¯)2=2 λ2β-2zR-2¯(λβ-1zR-1¯)2.
σmn2(z, λ)=γmnσ2(z, λ),
θmn2=γmnθ2,
σwmnθmn=γmnσwθ,
Mmn2=σwmnθmnλ/2π=γmnM2=γmn,
σwmnθmn¯λ¯/2π=σwsmnθsmn¯λ¯s/2π=γmn,
MPsmn2=γmnMPmn2θmn2β¯ βPoθmn2¯,1/2,
σmn4(z, λ)=ϵmn[σ2(z, λ)]2,
σmn4(z, λ)--r4|umn(x, y, z, λ)|2dxdy,
Kmn=ϵmn 1Pmn 0ρmn(λ)[σ2(z, λ)]2dλγmn21Pmn 0ρmn(λ)σ2(z, λ)dλ2,
Kmn=ϵmnγmn2+ϵmnγmn2(λ-λ¯)2(λ¯)2¯.
Kmnϵmnγmn2 λ2zR2¯(λzR¯)2.
Kmn=ϵmnγmn2 λ2zR-2¯(λzR-1¯)2.
βPo=1+zw2+zR2zR21F2-1F121+2zwF1+zw2+zR2F2,
1Fi=1λ¯P 0 ρ(λ)λfi dλ,i=1, 2.
θs2¯βPo-1θ2¯; zRPsβPozRP.
|umn(x, y, z, λ)|2=Hn22xw(z, λ)Hm22yw(z, λ)exp-2 x2+y2w(z, λ)22m+n-1m!n!πw(z, λ)2,
Hk(ξ)=(-1)k exp(ξ2) dk[exp(-ξ2)]dξk,k=0, 1, 2 ,
σmn4(z, λ)--r4|umn(x, y, z, λ)|2dxdy.
-ξ2Hk2(ξ)exp(-ξ2)dξ=2k-1k!π(2k+1),
-ξ4Hk2(ξ)exp(-ξ2)dξ=2k-23k!π(2k2+2k+1),
σmn4(z, λ)=ϵmn[σ2(z, λ)]2,
ϵmn=(m+n+1)2+m(m+1)2+n(n+1)2+1.
γmn=m+n+1.
ϵmnγmn2=1+2+m(m+1)+n(n+1)2(m+n+1)2.
limm,n fixed ϵmnγmn2=limn,m fixed ϵmnγmn2=1.50.
limm=n ϵmnγmn2=1.25.
|umn(r, z, λ)|2=1n!(n+m)! 1πr2 2r2w(z, λ)2m+1×Lnm2r2w(z, λ)22 exp-2 r2w(z, λ)2.
Lnm(ξ)=exp(ξ)ξm dn[exp(-ξ)ξ(n+m)]dξn,n, m=0, 1, 2,.
σmn4(z, λ)0r4|umn(r, z, λ)|2rdr.
0ξm+2[Lnm(ξ)]2 exp(-ξ)dξ=32(2n+m+1)2+(1+m)(1-m)2,
σmn4(z, λ)=ϵmn[σ2(z, λ)]2,
ϵmn=32(2n+m+1)2+(1+m)(1-m)2.
γmn=2n+m+1.
ϵmnγmn2=32+12 (1+m)(1-m)(2n+m+1)2.
limm,n fixed ϵmnγmn2=1.00.
limm,m fixed ϵmnγmn2=1.50.
limm=n ϵmnγmn2=1.44.

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